RU-239-OPEN 2015

Let the incircle of triangle $ABC$ touches the sides $AB,BC,CA$ in $C_1,A_1,B_1$ respectively. If $A_1C_1$ cuts the parallel to $BC$ from $A$ at $K$ prove that $\angle KB_1A_1=90.$

Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$ .

The edges of a graph $G$ are coloured in two colours. Such that for each colour all the connected components of this graph formed by edges of this colour contains at most $n>1$ vertices. Prove there exists a proper colouring for the vertices of this graph with $n$ colours.